Sierpiński graphs and Sierpiński triangle graphs form two-parametric families of connected simple graphs which are related, for , to the Tower of Hanoi with discs and for to the Sierpiński triangle fractal. The vertices of minimal degree play a special role as extreme vertices in and primitive vertices in . The key concept of this note is that of an -key vertex whose distance to one of the extreme or primitive vertices, respectively, is times the distance to another one. The number of such vertices and the distances occurring lead to integer sequences with respect to parameter like, e.g., the Fibonacci sequence (golden) for and the Pell sequence (silver) for . The elements of most of these sequences form self-generating sets. We discuss the cases in detail.